Central force


In classical mechanics, a central force on an object is a force that is directed along the line joining the object and the origin:
where is the force, F is a vector valued force function, F is a scalar valued force function, r is the position vector, ||r|| is its length, and = r/||r|| is the corresponding unit vector.
Not all central force fields are conservative or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric.

Properties

Central forces that are conservative can always be expressed as the negative gradient of a potential energy:-
.
In a conservative field, the total mechanical energy is conserved:
, and in a central force field, so is the angular momentum:
because the torque exerted by the force is zero. As a consequence, the body moves on the plane perpendicular to the angular momentum vector and containing the origin, and obeys Kepler's second law.
It can also be shown that an object that moves under the influence of any central force obeys Kepler's second law. However, the first and third laws depend on the inverse-square nature of Newton's law of universal gravitation and do not hold in general for other central forces.
As a consequence of being conservative, these specific central force fields are irrotational, that is, its curl is zero, except at the origin:

Examples

Gravitational force and Coulomb force are two familiar examples with being proportional to 1/r2 only. An object in such a force field with negative obeys Kepler's laws of planetary motion.
The force field of a spatial harmonic oscillator is central with proportional to r only and negative.
By Bertrand's theorem, these two, and , are the only possible central force fields where all bounded orbits are stable closed orbits. However, there exist other force fields, which have some closed orbits.